AR, MA and ARMA models

40 Pages·2016·0.29 MB·English

AR, MA and ARMA models
Stationarity 1 Stationarity
ACF
2 ACF
Ljung-Box
test
3 Ljung-Box test
Whitenoise
ARmodels 4 White noise
Example
5 AR models
PACF
AIC/BIC 6 Example
Forecasting
7 PACF
MAmodels
Summary 8 AIC/BIC
9 Forecasting
10 MA models
11 Summary
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Linear Time Series Analysis
Stationarity and Its Applications1
ACF
Ljung-Box
test For basic concepts of linear time series analysis see
Whitenoise
• Box, Jenkins, and Reinsel (1994, Chapters 2-3), and
ARmodels
• Brockwell and Davis (1996, Chapters 1-3)
Example
PACF
AIC/BIC
The theories of linear time series discussed include
Forecasting
MAmodels • stationarity
Summary
• dynamic dependence
• autocorrelation function
• modeling
• forecasting
1Tsay (2010), Chapter 2.
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Stationarity
ACF
The econometric models introduced include
Ljung-Box
test
Whitenoise (a) simple autoregressive models,
ARmodels
(b) simple moving-average models,
Example
(b) mixed autoregressive moving-average models,
PACF
(c) seasonal models,
AIC/BIC
(d) unit-root nonstationarity,
Forecasting
(e) regression models with time series errors, and
MAmodels
(f) fractionally differenced models for long-range
Summary
dependence.
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Strict stationarity
Stationarity
ACF
Ljung-Box The foundation of time series analysis is stationarity.
test
Whitenoise
ARmodels A time series {rt} is said to be strictly stationary if the joint
Example distribution of (r ,...,r ) is identical to that of
t1 tk
PACF (r ,...,r ) for all t, where k is an arbitrary positive
t1+t tk+t
AIC/BIC integer and (t ,...,t ) is a collection of k positive integers.
1 k
Forecasting
MAmodels
The joint distribution of (r ,...,r ) is invariant under time
Summary t1 tk
shift.
This is a very strong condition that is hard to verify
empirically.
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Weak stationarity
Stationarity
A time series {r } is weakly stationary if both the mean of r
ACF t t
and the covariance between r and r are time invariant,
Ljung-Box t t−l
test where l is an arbitrary integer.
Whitenoise
ARmodels
Example More specifically, {rt} is weakly stationary if
PACF (a) E(rt) = µ, which is a constant, and
AIC/BIC (b) Cov(r ,r ) = γ , which only depends on l.
t t−l l
Forecasting
MAmodels In practice, suppose that we have observed T data points
Summary {r |t = 1,...,T}. The weak stationarity implies that the
t
time plot of the data would show that the T values fluctuate
with constant variation around a fixed level.
In applications, weak stationarity enables one to make
inference concerning future observations (e.g., prediction).
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Properties
Stationarity
ACF The covariance
Ljung-Box γ = Cov(rt,rt−l)
test
Whitenoise is called the lag-l autocovariance of r .
t
ARmodels
Example
It has two important properties:
PACF
AIC/BIC (a) γ0 = Var(rt), and (b) γ−l = γl.
Forecasting
MAmodels
The second property holds because
Summary
Cov(r ,r ) = Cov(r ,r )
t t−(−l) t−(−l) t
= Cov(r ,r )
t+l t
= Cov(r ,r −l),
t1 t1
where t = t+l.
1
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Autocorrelation function
Stationarity
ACF
Ljung-Box The autocorrelation function of lag l is
test
Whitenoise Cov(r ,r ) Cov(r ,r ) γ
t t−l t t−l l
ρ = = =
ARmodels l (cid:112)Var(rt)Var(rt−l) Var(rt) γ0
Example
PACF where the property Var(r ) = Var(r ) for a weakly
t t−l
AIC/BIC stationary series is used.
Forecasting
MAmodels
Summary In general, the lag-l sample autocorrelation of r is defined as
t
(cid:80)T (r −r¯)(r −r¯)
ρˆ = t=l+1 t t−l 0 ≤ l < T −1.
l (cid:80)T (r −r¯)2
t=1 t
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Portmanteau test
Stationarity
ACF
Box and Pierce (1970) propose the Portmanteau statistic
Ljung-Box
test
m
Whitenoise Q∗(m) = T (cid:88)ρˆ2
ARmodels l
l=1
Example
PACF as a test statistic for the null hypothesis
AIC/BIC
Forecasting H : ρ = ··· = ρ = 0
0 1 m
MAmodels
Summary against the alternative hypothesis
H : ρ (cid:54)= 0 for some i ∈ {1,...,m}.
a i
Under the assumption that {r } is an iid sequence with
t
certain moment conditions, Q∗(m) is asymptotically χ2 .
m
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Ljung and Box (1978)
Stationarity
ACF
Ljung-Box
test
Ljung and Box (1978) modify the Q∗(m) statistic as below
Whitenoise
ARmodels to increase the power of the test in finite samples,
Example
PACF Q(m) = T(T +2)(cid:88)m ρˆ2l .
AIC/BIC T −l
l=1
Forecasting
MAmodels
Summary
The decision rule is to reject H if Q(m) > q2, where q2
0 α α
denotes the 100(1−α)th percentile of a χ2 distribution.
m
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# Load data
Stationarity da = read.table("http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts3/m-ibm3dx2608.txt",
header=TRUE)
ACF
# IBM simple returns and squared returns
Ljung-Box sibm=da[,2]
test sibm2 = sibm^2
Whitenoise # ACF
par(mfrow=c(1,2))
ARmodels acf(sibm)
Example acf(sibm2)
PACF # Ljung-Box statistic Q(30)
Box.test(sibm,lag=30,type="Ljung")
AIC/BIC Box.test(sibm2,lag=30,type="Ljung")
Forecasting > Box.test(sibm,lag=30,type="Ljung")
MAmodels Box-Ljung test
Summary
data: sibm
X-squared = 38.241, df = 30, p-value = 0.1437
> Box.test(sibm2,lag=30,type="Ljung")
Box-Ljung test
data: sibm2
X-squared = 182.12, df = 30, p-value < 2.2e-16
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